Questions tagged [profile-likelihood]

The profile likelihood function is an inference function constructed from the likelihood function. If the likelihood function depends on many parameters and only some are of interest, then the other parameters are removed "by concentration", which means that they are "maxed out".

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29 views

Standard errors of parameters in hyperparameter tuning

I have a model with the parameters $\textbf{θ} = (\textbf{φ}, ψ)$, which I trying to estimate. There is an out-of-the-box solution for ML estimating $\textbf{φ}$ if $ψ$ is fixed. If I treat $ψ$ as ...
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Errors on 'Profiling Confidence Interval' after Maximum Likelihood Estimation in R

I'm practicing a cancer treatment model using Maximum Likelihood Estimation with 'bbmle' packages in R. Step 1. I generated a virtual dataset ...
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Estimating the denominator of a binomial probability

Let $X \sim \mathrm{Binomial}\left(n, \frac{a^*}{b^*} \right)$, where $a^*$ and $b^*$ are coprime integers and $0 < a^* < b^* < 1000$. Here $n$ is known, but $a^*$ and $b^*$ are unknown. What'...
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Profile Likelihood confidence interval

I am interested in obtaining profile likelihood confidence intervals for parameter identifiability. My cost function is the least square error between the data and some fitted approximation depending ...
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Compare which of two groups is MORE similar to a third group?

I know I can use an ANOVA (+ post-hoc tests) to determine the equality (or inequality) of three groups: (i.e., H0: each of 3 means are equal; H A: at least 1 of 3 means is not equal). However, how ...
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error code in finding confidence interval

I optimized parameters from a function using mle2 from bbmle package. After I got the values, I tried to find profile likelihood confidence interval of parameters using confint() but there is one ...
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Estimating the noise to estimate profile likelihood

I am trying to obtain profile likelihoods around parameters obtained by fitting an ODE model to some data. I am using the method discussed in the study Data is assumed to have normal errors ~$N(0,\...
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247 views

Profile likelihood confidence interval proof

I have read that for null hypothesis $H_0 : \beta = \beta_0$, the likelihood ratio statistic from profile-likelihood is $$LR = 2 (\log L_p(\hat{\beta_p}) - \log L_p(\beta_0))$$ where $\hat{\beta_p}$ ...
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How to practically calculate confidence interval from the following confidence equation?

How to use the following equation to calculate the confidence interval: $$CI_{j,\alpha}(y) = \left\{p | -2PL_j(p) \leq \min_\theta - 2LL(y|\theta) + \Delta(\alpha) \right\}.$$ Here, CI is confidence ...
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How is to maximize a function $f(x,y)$ for values of $y$?

Maximum Likelihood Method: Likelihood function asks what value of parameter, $\theta$, makes the data set most probable. Let the distribution is $$f(x;p)=\binom{3}{x}p^x(1-p)^{n-x},\quad x=0,1,2,3.$$ ...
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Profile Likelihood: why optimize all other parameters while tracing a profile for a partitcular one?

Profile likelihood is sometimes used to get estimates for the confidence limits of parameters from an n-dimension parameter fit to a model. It can be used for example instead of Monte Carlo estimation....
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MLE: Marginal vs Full Likelihood

Suppose I have a statistical model with parameters $\boldsymbol{\theta}=\{\theta_1,\theta_2,\dots,\theta_n\}$ of which only a single parameter, say $\theta_1$, is of interest to me. Suppose also that ...
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Profile log-likelihood function empirical similarity

I´m trying to deduce the profile log-likelihood function in "Rule-Based and Case-Based Reasoning in Housing Prices" (page 14) by Gayer et al. (2007). Given is: $$\mathbf{y}=\alpha \mathbf{1} + \...
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618 views

Confidence intervals for GLMM: bootstrap vs likelihood profile

I've built a relatively large negative binomial GLMM (~50000 observations, 10 covariates in conditional model, 1 covariate as zero-inflation model, 3 random intercepts (650, 26, 26 levels in each ...
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Is the profile likelihood (dependent on one parameter) always a concave function?

Let $\theta = (\theta_1, \theta_2, \dots)$ be a vector of parameters, and let $L(\theta) = L(\theta|y)$ denote the likelihood function with respect to observed data $y$. Following the notation from ...
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123 views

Confidence interval on the percentage difference of two binomial distributions

I have survey data for women and men following a binomial distribution. Their means are $p_1$ and $p_2$ respectively. I have calculated $(p_1-p_2)/p_2$ and would like to attach a confidence interval ...
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Is my explanation of profile likelihood plots correct?

Using the metafor package in R to conduct a mixed-effects meta-analysis and meta-regression, I checked the profile likelihood ...
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248 views

Confidence intervals with penalized likelihood

I am trying to perform parameter estimation using something like a maximum likelihood ratio method, however I need to add a penalty term to constrain nuisance parameters which describe certain ...
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682 views

Firth's method for logistic regression - interpretation of R output

I have a multivariate, multinomial logistic regression model with exclusively continuous covariates. After some examination, I found that I had a problem of quasi-complete separation. The textbook ...
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875 views

Existence and uniqueness of MLE

I am aiming to show that the MLE of $(\alpha, \lambda)$ for $X_1, \dots, X_n \sim \Gamma(\alpha,\lambda)$ exists and is unique, under the assumption that the $X_i$ are all positive and unequal. I am ...
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240 views

Can we calculate the MLE of $\mu$ and $\sigma^2$ of normally distributed data using the profile likelihood approach?

My definition of profile likelihood is that given a vector of parameters $(\theta_1, \theta_2)$, with $\theta_1$ the parameter of interest, and $\theta_2$ a nuisance parameter -- If $L(\theta_1, \...
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Calculating the amount of underestimation

Coverage rate for a parameter is 91.2%, and the nominal coverage rate is 95%. If the confidence interval is based on asymptotic standard normal, then the amount of coverage 91.2% implies that the ...
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152 views

Restricted Maximum Likelihood Estimation for Linear Mixed Model

The maximum likelihood estimation procedure for linear mixed model is described in this link. It seems to me that something is wrong there. In their Restricted Maximum Likelihood section the first ...
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287 views

Profile likelihood

I am considering a normal distribution with mean $\beta_1 + \beta_2\exp(-\phi x)$ and variance $\sigma^2$, i.e. $y \sim N(\beta_1 + \beta_2\exp(-\phi x), \sigma^2) $. My aim is to calculate the ...
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Discordant significance of OR’s confidence interval and p-value in glm(quasibinomial) model [R]

I’m currently trying to test if differences in proportions of people infected by malaria (RDT positive) between clusters with high or low coverage of control intervention are significant. Therefore I’...
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272 views

Determining if two profile likelihood curves are significantly different

I want to compare two profile likelihood curves and determine if they are significantly different from one another. For example are the following curves significantly different from one another: I ...
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Parameter distribution from profile likelihood

This question is regarding profile likelihood to obtain CI of parameters. There is a published parameter values that best fits (using minimum chi square)the data for some mathematical model say y=f(...
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Profile likelihood confidence intervals = (-Inf Inf) in a glmer {lme4_1.1-7}

I fitted a mixed logit model with crossed random effects in lme4_1.1-7::glmer (R version 3.1.1 / OS X 10.9.4 Mavericks). Had to simplify the maximal random-effect ...
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How to calculate confidence intervals in a GLM using the profile likelihood?

I've been trying to better understand how JMP does regression and associated models. I can compute the correct parameter estimates for a GLM, by using iteratively re weighted least squares. But now I'...
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655 views

How can be profile plots in EVT interpreted and what is the theoretical nature of it?

I have two question about profile log-likelihood. One is of theoretical nature and one is about a plot in R. I illustrate the two questions in a application of profile-likelihood in EVT. I use the ...
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497 views

Inconsistency in Two stage Maximum Likelihood Estimation

I want to maximize a log-likelihood function (L) that is a function of parameters $\beta_i$ for $i=1,..,k$ and $\alpha_1, \alpha_2$. Ideally, I want to perform the estimation of all parameters in one ...
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Confidence Interval for Inverse Gamma Distribution

I would like to understand if there exists any method to find confidence interval for the parameters of inverse gamma distribution.
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SAS profile-likelihood confidence intervals and p-values - why is there a discrepancy?

Using Firth's penalized likelihood method in logistic regression, why can you get a p-value of <.05 when the confidence interval for the estimate includes 1? code: ...
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Yet another “Bayesian vs Maximum Likelihood” question

In the fully Bayesian approach, the predictive distribution is: $$ P( Y|X ) = \int P(\theta | X ) P( Y | \theta ) d\theta $$ When the integral is difficult to compute, we might resort to the Maximum ...
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How to use profile likelihood?

I am new to profile likelihood and do not really understand what advantages it may have. Lets say I have the following results estimating the means of three groups. What can I say about them? R code: ...
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What is the relationship between profile likelihood and confidence intervals?

To make this chart I generated random samples of different size from a normal distribution with mean=0 and sd=1. Confidence intervals were then calculated using alpha cutoffs ranging from .001 to .999 ...
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Is this a nested model and can I use the likelihood ratio test?

I have a data set, total_data and I applied a model to it. For instance, the model has one parameter $\beta$, and I calculated the log-likelihood of the fitted model (using maximum likelihood method). ...
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114 views

Manipulating Binomial Distribution

Recently, I've been reading Yudi Pawitan's book, In All Likelihood. In the book, there's a section on profile likelihood; the methods explored in this section are subsequently applied to some data on ...
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472 views

Likelihood analysis for exponential distribution

Assume a collection of independent exponential random variables $y_{1}, \ldots, y_{n}$ with means $\mu_{1}, \ldots, \mu_{n}$; where $\mu_{i} = \beta_{0}+\beta_{1}x_{i}$. How can I find the profile ...
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Profile likelihood confidence intervals

The theory behind profile likelihood (PL) confidence intervals (CIs) is clear to me. (See here, for example). SAS is surprisingly quick in calculating the PL CIs for all the covariates in a given ...
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Help me figure out the profile likehihood given this covariance function

I'm taking a spatial stats class, and I'm on the road so I can't ask the prof for help. Would appreciate help understanding what is going on here. The problem is set up with $Y = X_s'\beta + e$ (...
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Possible outcomes of approximate profile-likelihood estimator (APLE) for spatial autocorrelation

I've been working with spatial autocorrelation for a while and now I'm trying to move from more traditional estimators such as Moran's I or Geary's C to the new APLE estimator. I read Li's papers on ...
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71 views

Possible outcomes of Approximate profile-likelihood estimator (APLE) for spatial autocorrelation

I've been working with spatial autocorrelation for a while and now I'm trying to move from more traditional estimators such as Moran's I or Geary's C to the new APLE estimator. I read Li's papers on ...
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91 views

References about profiled deviance components

I have read the excellent slides "Assessing the precision of estimates of variance components" from Douglas Bates. I'm looking for some references about these profiled deviance components (for ...
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Who invented profile maximum likelihood estimation?

Could anyone give me some information on who invented profile maximum likelihood estimation or who first use profile maximum likelihood estimation and the short history of profile maximum likelihood ...
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How can I estimate 95% confidence intervals using profiling for parameters estimated by maximising a log-likelihood function using optim in R?

How can I estimate 95% confidence intervals using profiling for parameters estimated by maximising a log-likelihood function using optim in R? I know I can asymptotically estimate the covariance ...
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What is the exact definition of profile likelihood?

Does anyone here know the exact definition of Profile Likelihood? Or does it have one?
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A little confusion on Profile likelihood

As we all know, profile likelihood is an effective method for the estimation of conditional parametric model. But I still don't know exactly why it works. Profile likelihood was thoroughly studied by ...
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469 views

Profile likelihood of N in binomial model

Let $(n_1,...,n_k)$ be a sample from a Binomial$(N,p)$ where both parameters are unknown. In many cases, the profile likelihood of $N$ is asymptotic in the sense that it never decays to $0$. An ...
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“On/off problem” resources

I need to study in depth the on/off problem with different approaches: frequentist, bayesian - frequentist hybrid, profile likelihood. The on/off problem is a counting experiment where you observe $...